Problem 31 - Elementary and Intermediate Algebra

Question

Write decimal notation for each number. $$ \frac{2}{3} $$

Step-by-Step Solution

Answer

0.666... or 0.\bar{6}

Step 1 - Understand the Fraction

Recognize that the given fraction is \(\frac{2}{3}\).

Step 2 - Perform Division

Divide the numerator (2) by the denominator (3) to convert the fraction into decimal form. \(2 \div 3 = 0.666\text{...}\)

Step 3 - Identify Repeating Decimal

Notice that the digits after the decimal repeat. This fraction can be written in decimal notation as \(\frac{2}{3} = 0.666\text{...} = 0.\bar{6}\).

Key Concepts

Fraction to Decimal

When you want to convert a fraction to a decimal, the process involves dividing the numerator by the denominator. This is a key mathematical concept that helps in understanding how parts of a whole can be represented in different forms. For example, in our exercise, we start with the fraction \(\frac{2}{3}\). Here, 2 is the numerator, and 3 is the denominator. By performing the division operation, we can convert this fraction into its decimal equivalent. Dividing 2 by 3 gives us 0.666..., with the number 6 repeating indefinitely. This simple division method works for any fraction, whether the result is a terminating or repeating decimal.

Repeating Decimal

A repeating decimal, also known as a recurring decimal, is a decimal number that has one or more repeating digits after the decimal point. In our example, \(\frac{2}{3} = 0.666...\), the digit 6 repeats. To indicate a repeating decimal, we place a bar over the repeating digit(s), so \(\frac{2}{3}\) becomes \(0.\bar{6}\). This notation helps us understand that the sequence of 6s continues infinitely without changing. Repeating decimals often arise from fractions where the denominator has factors other than 2 or 5. Identifying and writing repeating decimals correctly ensures clarity in mathematical communication and understanding.

Division Method

The division method is used to convert fractions into decimals. Here's how it works in a step-by-step manner:

• Step 1: Identify the numerator and denominator in the fraction. Here, for \(\frac{2}{3}\), 2 is the numerator, and 3 is the denominator.
• Step 2: Perform the division operation going step by step. Divide 2 by 3, which cannot be done directly. So, add a decimal point and zeros to the numerator, making it 2.0 or 20 tenths.
• Step 3: Divide 20 by 3, which goes 6 times (since 3*6=18). Write down 6 after the decimal point.
• Step 4: Subtract 18 from 20, leaving a remainder of 2. Bring down another 0, making it 20 again, and repeat the division step creating an ongoing process.

The continuous quotient of 6 shows the repeating nature of the division, giving us the repeating decimal \(0.\bar{6}\). This method is fundamental and widely used in mathematics to understand the relationship between fractions and their equivalent decimal forms.